Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras

Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2007) Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras. [MIMS Preprint]

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Abstract

Given a class of structured matrices $\Sb$, we identify pairs of vectors $x,b$ for which there exists a matrix $A\in\Sb$ such that $Ax=b$, and also characterize the set of all matrices $A\in\Sb$ mapping $x$ to $b$. The structured classes we consider are the Lie and Jordan algebras associated with orthosymmetric scalar products. These include (skew-)symmetric, (skew-)Hamiltonian, pseudo (skew-)Hermitian, persymmetric and perskew-symmetric matrices. Structured mappings with extremal properties are also investigated. In particular, structured mappings of minimal rank are identified and shown to be unique when rank-1 is achieved. The structured mapping of minimal Frobenius norm is always unique and explicit formulas for it and its norm are obtained. Finally the set of all structured mappings of minimal 2-norm is characterized. Our results generalize and unify existing work, answer a number of open questions, and provide useful tools for structured backward error investigations.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	Lie algebra, Jordan algebra, scalar product, bilinear form, sesquilinear form, orthosymmetric, adjoint, structured matrix, backward error, Hamiltonian, skew-Hamiltonian, Hermitian, complex symmetric, skew-symmetric, persymmetric, perskew-symmetric, minimal rank, minimal Frobenius norm, minimal 2-norm.
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Dr Françoise Tisseur
Date Deposited:	12 Jun 2007
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/230

Available Versions of this Item

• Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras. (deposited 27 Mar 2006)
  • Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras. (deposited 21 Apr 2006)
    • Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras. (deposited 12 Jun 2007) [Currently Displayed]

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